Let $X$ and $Y$ be two nonsingular projective varieties defined over the complex numbers.
If $X$ and $Y$ are $\mathbb{A}^1$-homotopy equivalent, then does $X$ satisfying the Hodge conjecture imply that $Y$ satisfies the Hodge conjecture?
Is the Hodge Conjecture an $\mathbb{A}^1$-homotopy invariant?
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